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Proceedings of the American Mathematical Society

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Arcs defined by one-parameter semigroups of operators

Authors: Hugo D. Junghenn and C. T. Taam
Journal: Proc. Amer. Math. Soc. 44 (1974), 113-120
MSC: Primary 47D05
MathSciNet review: 0331114
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Abstract: Let $ T(t)(t \geqq 0)$ be a one-parameter semigroup of continuous linear operators in a locally convex reflexive linear topological space $ X$ such that $ T(c)$ is an isomorphism (into) for some $ c > 0$. It is proved that for any $ x \in X,T( \cdot )x$ is of bounded variation on finite intervals if and only if $ x$ is in the domain of the infinitesimal generator of $ T(t)$. The result is interpreted geometrically in terms of arc-length.

References [Enhancements On Off] (What's this?)

  • [1] P. L. Butzer and H. Berens, Semi-groups of operators and approximation, Die Grundlehren der math. Wissenschaften, Band 145, Springer-Verlag, New York, 1967. MR 37 #5588. MR 0230022 (37:5588)
  • [2] T. Kōmura, Semigroups of operators in locally convex spaces, J. Functional Analysis 2 (1968), 258-296. MR 38 #2634. MR 0234317 (38:2634)

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Keywords: Semigroup of operators, reflexive, arc-length, bounded variation, absolute continuity, local equicontinuity
Article copyright: © Copyright 1974 American Mathematical Society

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